A003992 - OEIS

A003992

Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

%I #50 Sep 08 2022 08:44:32

%S 1,1,0,1,1,0,1,2,1,0,1,3,4,1,0,1,4,9,8,1,0,1,5,16,27,16,1,0,1,6,25,64,

%T 81,32,1,0,1,7,36,125,256,243,64,1,0,1,8,49,216,625,1024,729,128,1,0,

%U 1,9,64,343,1296,3125,4096,2187,256,1,0,1,10,81,512,2401,7776,15625,16384,6561,512,1,0

%N Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

%C If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - _Robert A. Russell_, Nov 08 2018

%C T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - _Geoffrey Critzer_, Apr 03 2020

%H T. D. Noe, <a href="/A003992/b003992.txt">Rows n = 0..50 of triangle, flattened</a>

%F E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - _Paul D. Hanna_, Oct 22 2004

%F E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - _Franklin T. Adams-Watters_, Jun 23 2006

%e Rows begin:

%e [1, 0, 0, 0, 0, 0, 0, 0, ...],

%e [1, 1, 1, 1, 1, 1, 1, 1, ...],

%e [1, 2, 4, 8, 16, 32, 64, 128, ...],

%e [1, 3, 9, 27, 81, 243, 729, 2187, ...],

%e [1, 4, 16, 64, 256, 1024, 4096, 16384, ...],

%e [1, 5, 25, 125, 625, 3125, 15625, 78125, ...],

%e [1, 6, 36, 216, 1296, 7776, 46656, 279936, ...],

%e [1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

%t Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

%o (PARI) T(n,k) = (n-k)^k \\ _Charles R Greathouse IV_, Feb 07 2017

%o (Magma) [[(n-k)^k: k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Nov 08 2018

%Y Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.

%Y Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.

%Y Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.

%Y Cf. A099555.

%Y Transpose is A004248. See A051128, A095884, A009999 for other versions.

%Y Cf. A277504 (unoriented), A293500 (chiral).

%K easy,nice,nonn,tabl

%O 0,8

%A _Marc LeBrun_

%E More terms from _David W. Wilson_

%E Edited by _Paul D. Hanna_, Oct 22 2004